Meshfree methods for the variable-order fractional advection–diffusion equation

Abstract

The fractional advection–diffusion equation can describe the anomalous diffusion associated with complicated diffusion medium and pollution source in application problems. The variable-order fractional advection–diffusion equation can change the order of fractional operator by controlling the power of order function. In this paper, 1D and 2D time–space variable-order fractional advection–diffusion equations are studied on a bounded domain, in which the order of time fractional derivatives are time dependent while the order of space fractional derivatives depend on either time or space. The time fractional derivatives are approximated by full-implicit difference scheme and the space fractional derivatives are discretized via Kansa’s method combined with Wendland’s C6 compactly supported radial basis function. The Gauss–Jacobi quadrature is utilized to evaluate the weakly singular integration during the computation of the space fractional derivative of radial basis function. Finally, some numerical experiments are provided to verify the accuracy and efficiency of the proposed methods in 1D and 2D cases.